论文标题
非Convex ADMM的收敛性，并应用于CT成像
Convergence for nonconvex ADMM, with applications to CT imaging
论文作者
论文摘要
乘数的交替方向方法（ADMM）算法是一种功能强大且灵活的工具，用于$ \ min \ {f（x）+g（y）形式的复杂优化问题：ax+by = c \} $。 ADMM在各种具有挑战性的环境中表现出强大的经验表现，包括非平滑度和目标函数的非概念$ f $ and $ g $，并为计算机断层扫描（CT）成像的图像重建的反向问题提供了一种简单而自然的方法。从理论的角度来看，在非convex设置中的收敛结果通常假定目标中至少一个组件函数中的平滑度。在这项工作中，我们的新理论结果在受限的强凸度假设下提供了融合保证，而无需平滑度或可不同，同时仍然允许在需要时近似处理可区分的术语。我们从经验上验证了这些理论结果，其中一个模拟示例，其中$ f $和$ g $都是非不同的，因此超出了现有理论的范围，以及模拟的CT图像重建问题。
The alternating direction method of multipliers (ADMM) algorithm is a powerful and flexible tool for complex optimization problems of the form $\min\{f(x)+g(y) : Ax+By=c\}$. ADMM exhibits robust empirical performance across a range of challenging settings including nonsmoothness and nonconvexity of the objective functions $f$ and $g$, and provides a simple and natural approach to the inverse problem of image reconstruction for computed tomography (CT) imaging. From the theoretical point of view, existing results for convergence in the nonconvex setting generally assume smoothness in at least one of the component functions in the objective. In this work, our new theoretical results provide convergence guarantees under a restricted strong convexity assumption without requiring smoothness or differentiability, while still allowing differentiable terms to be treated approximately if needed. We validate these theoretical results empirically, with a simulated example where both $f$ and $g$ are nondifferentiable -- and thus outside the scope of existing theory -- as well as a simulated CT image reconstruction problem.
